Invertible Darboux Transformations

Shemyakova, Ekaterina

doi:10.3842/sigma.2013.002

Cited by 11 publications

(12 citation statements)

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“…Note that one may also view the ∼ classes as being the Darboux transformations, rather than the individual pairs of operators. This approach was taken in [31,32,33] and [34].…”

Section: Types Of Darboux Transformationsmentioning

confidence: 99%

Classification of Multidimensional Darboux Transformations: First Order and Continued Type

Hobby¹,

Shemyakova²

2017

SIGMA

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Abstract. We analyze Darboux transformations in very general settings for multidimensional linear partial differential operators. We consider all known types of Darboux transformations, and present a new type. We obtain a full classification of all operators that admit Wronskian type Darboux transformations of first order and a complete description of all possible first-order Darboux transformations. We introduce a large class of invertible Darboux transformations of higher order, which we call Darboux transformations of continued Type I. This generalizes the class of Darboux transformations of Type I, which was previously introduced. There is also a modification of this type of Darboux transformations, continued Wronskian type, which generalize Wronskian type Darboux transformations.

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“…Note that one may also view the ∼ classes as being the Darboux transformations, rather than the individual pairs of operators. This approach was taken in [31,32,33] and [34].…”

Section: Types Of Darboux Transformationsmentioning

confidence: 99%

Classification of Multidimensional Darboux Transformations: First Order and Continued Type

Hobby¹,

Shemyakova²

2017

SIGMA

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“…The first steps in studying invertible Darboux trans formations were made in [8], where necessary condi tions of invertibility for some first order Darboux transformations were obtained. In this paper, a general algebraic (categorial) approach put forward in [9, Sec tion 3] is developed.…”

Section: Introductionmentioning

confidence: 99%

Invertible Darboux transformations of type I

Shemyakova

2015

Program Comput Soft

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In the paper, a general category based approach to invertible Darboux transformations is sug gested. For operators of arbitrary dimension and arbitrary order, it is suggested to consider Darboux transfor mations of certain type, namely, of type I. Explicit compact formulas are obtained. In particular, any invert ible Darboux transformations of first order and, for example, Laplace transformations are type I transforma tions. For operators of third order in the plane, analogues of the Darboux transformation chains are obtained.

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“…Further classes of generalization include Darboux transformations of type I [17,18,19] and of continued type [9]. A looser and more general notion of intertwining Laplace transformation than that presented in this paper can be found in [7].…”

Section: Introductionmentioning

confidence: 99%